Two sound sources are moving in opposite directions with velocities ` v_(1) and v_(2) ( v_(1) gt v_(2))` . Both are moving away from a stationary observer. The frequency of both the sources is 900 Hz. What is the value of ` v_(1) - v_(2)` so that the beat frequency aboserved by the observer is 6 Hz. speed of sound v= 300 m/s given ,that `v_(1) and v_(2) lt lt v `

Options

A. 1m/s
B. 2m/s
C. 3m/s
D. 4 m/s

Answer

To solve the problem step by step, we need to find the value of \( v_1 - v_2 \) such that the beat frequency observed by the stationary observer is 6 Hz. We know the following:

- Frequency of both sources, \( f = 900 \) Hz
- Speed of sound, \( v = 300 \) m/s
- Beat frequency, \( f_b = 6 \) Hz
- \( v_1 > v_2 \) and both are much less than \( v \).

### Step 1: Write the formula for the observed frequency
The formula for the frequency observed by a stationary observer when the source is moving away is given by:

\[ f' = \frac{f \cdot v}{v + v_s} \]

where:
- \( f' \) is the observed frequency,
- \( f \) is the source frequency,
- \( v \) is the speed of sound,
- \( v_s \) is the speed of the source.

### Step 2: Calculate the observed frequencies for both sources
For the first source moving with speed \( v_1 \):

\[ f_1' = \frac{900 \cdot 300}{300 + v_1} \]

For the second source moving with speed \( v_2 \):

\[ f_2' = \frac{900 \cdot 300}{300 + v_2} \]

### Step 3: Simplify the observed frequencies
Using the approximation \( v_1, v_2 \ll v \), we can expand the observed frequencies using a binomial approximation:

\[ f_1' \approx 900 \left( 1 - \frac{v_1}{300} \right) \]
\[ f_2' \approx 900 \left( 1 - \frac{v_2}{300} \right) \]

### Step 4: Find the beat frequency
The beat frequency \( f_b \) is given by the absolute difference of the two observed frequencies:

\[ f_b = |f_2' - f_1'| = |(900 - 3v_2) - (900 - 3v_1)| \]

This simplifies to:

\[ f_b = |3v_1 - 3v_2| = 3|v_1 - v_2| \]

### Step 5: Set the beat frequency equal to 6 Hz
According to the problem, the beat frequency is 6 Hz:

\[ 3|v_1 - v_2| = 6 \]

### Step 6: Solve for \( v_1 - v_2 \)
Dividing both sides by 3 gives:

\[ |v_1 - v_2| = 2 \]

Since \( v_1 > v_2 \), we have:

\[ v_1 - v_2 = 2 \text{ m/s} \]

### Final Answer
Thus, the value of \( v_1 - v_2 \) is \( 2 \) m/s.

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